| |Extremal metric has been a hot point in differential geometry since 1982. It is a natural generalization of the famous Kaehler-Einstein metric. An HCMU metric is a kind of extremal metric with singularities on a compact Riemann surface. In this paper we prove that any HCMU metric corresponds to a meromorphic 1-form (called character 1-form) with some properties, then we prove that the existence of HCMU metric with given singular data is equivalent to the existence of character 1-form.
The paper is in Math. Ann., Vol.351, issue 2(2011), 327-345.